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Proceedings of the Boyal Society 
manner of example (2.), and having cyclic irrotational motion of 
the liquid through its perforations, is a case of steady motion 
To this case belongs the irrotational motion of liquid in the neigh 
bourhood of any rotationally moving portion of fluid of the same 
shape as the solid, provided the distribution of the rotational mo 
tion is such that the shape of the portion endowed with it remains 
unchanged. The object of the present paper is to investigate 
general conditions for the fulfilment of this proviso ; and to inves 
tigate, farther, the conditions of stability of distribution of vortex 
motion satisfying the condition of steadiness. 
3. General synthetical condition for steadiness of vortex motion . — 
The change of the fluid’s molecular rotation at any point fixed in 
space must be the same as if for the rotationally moving portion 
of the fluid were substituted a solid, with the amount and direction 
of axis of the fluid’s actual molecular rotation inscribed or marked 
at every point of it, and the whole solid, carrying these inscrip- 
tions with it, were compelled to move in some manner answering 
to the description of example (2). If at any instant the distribu- 
tion of molecular rotation * through the fluid, and corresponding 
distribution of fluid velocity, are such as to fulfil this condition, it 
will be fulfilled through all time. 
4. General analytical condition for steadiness of vortex motion . — 
If, with (§ 24, below) vorticity and “ impulse,” given, the kinetic 
energy is a maximum or a minimum, it is obvious that the motion 
is not only steady, but stable. If, with same conditions, the 
energy is a maximum-minimum, the motion is clearly steady, but 
it may be either unstable or stable. 
5. The simple circular Helmholtz ring is a case of stable steady 
motion, with energy maximum-minimum for given vorticity and 
given impulse. A circular vortex ring, with an inner irrotational 
annular core, surrounded by a rotationally moving annular shell 
(or endless tube), with irrotational circulation outside all, is a case 
of motion which is steady, if the outer and inner contours of the 
* One of the Helmholtz’s now well-known fundamental theorems shows 
that, from the molecular rotation at every point of an infinite fluid the velocity at 
every point is determinate, being expressed synthetically by the same formulae 
as those for finding the “ magnetic resultant force” of a pure electro-magnet. 
■ — Thomson’s Reprint of Papers on Electrostatics and Magnetism. 
